Another major issue deserves special mention. Correcting for biases
that may be present in the data or data samples is a topic for the
statistics of data analysis which extends beyond the scope of this
paper. The problems associated with sample statistics are complex and
involved, more so for some distance indicators than for others, although
all indicators are affected to some degree.
Unfortunately, there is no single prescription that is appropriate in all
circumstances, and a complete discussion is better addressed in another
review
(Bertschinger 1992).
For the purposes of this paper, we call
the attention of the reader to a few common biases that afflict several
techniques, and these should be kept in mind throughout this review.

A true
Malmquist (1920)
effect may be present such that any sample of
objects becomes more and more restricted to brighter members as
distance increases. Thus, a sample of distant galaxies will have a higher
average luminosity than a nearby sample. This effect, in itself, does
not mandate biased distances provided that any parameter used to
predict the galaxy luminosity is not also biased.

A more subtle effect can arise if the galaxy sample is chosen from a set of
biased parameters and those parameters are also used to estimate distances
(e.g., magnitudes or diameters taken from a photographic catalog).
If those same parameters, which have some measurement uncertainty, enter
into the calculation of distances,
then the sample will include galaxies near the selection limit
that should have been
rejected (i.e., they appear brighter or larger than the selection limit).
The improper sample selection introduces biased distance estimates near
the completeness limit which, in turn, biases the sample mean.

Biases can also result from differences in the environments of the samples.
For example, most methods are calibrated locally but applied in large
galaxy clusters. If the galaxies in clusters are somehow different
from those in the Local Group, a very simple bias arises. Similarly,
first-ranked galaxies may foster unusual objects (e.g., extremely bright
planetary nebulae, globular clusters, red giants) which do not develop
in the Local Group, again leading to a simple calibration bias.

Another subtle form of bias can be introduced in the analysis phase of
distance determination by binning data. A histogram of globular
clusters or planetary nebulae, for example, will have a bias which
scatters objects from heavily populated bins into bins with fewer
objects. Proper analysis techniques (e.g., maximum likelihood, numerical
simulations) can account for this effect, first discussed by
Eddington
(1913,
1940),
and more recently by
Trumpler and Weaver
(1962).
A variant of the problem is
discussed by
Lynden-Bell et
al. (1988)
in the context of deriving the
average distance to a cluster of galaxies. When the distance measuring
technique has an uncertainty which is large enough that a significant number of
background and foreground galaxies are inadvertently included in
the sample, the mean distance for the cluster sample may be biased, depending
on the spatial density of the sample and interloper galaxies.